For millimeter wave (mmWave) or Terahertz (THz) communications, by leveraging the high spatial resolution offered by large antenna arrays and the multi-path sparsity of mmWave/THz channels, a novel inter-symbol interference (ISI) mitigation technique called delay alignment modulation (DAM) has been recently proposed. The key ideas of DAM are delay pre-compensation and path-based beamforming. However, existing research on DAM is mainly based on fully digital beamforming, which requires the number of radio frequency (RF) chains to be equal to the number of antennas. This paper proposes the hybrid analog/digital beamforming based DAM, including both fully and partially connected structures. The analog and digital beamforming matrices are designed to achieve performance close to DAM based on fully digital beamforming. While DAM was considered for the path-based channel model with integer delays in the previous work, this paper extends DAM to a more general tap-based model that accounts for fractional path delays. To further reduce the cost of channel estimation and improve the performance for wireless channels with fractional delays, DAM with codebook-based beam alignment and DAM-orthogonal frequency division multiplexing (DAM-OFDM) with hybrid beamforming are proposed. The effectiveness of the proposed techniques is verified by extensive simulation results.